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# If $\Delta = \begin{vmatrix} a_11&a_12&a_13 \\ a_21&a_22&a_23 \\ a_31&a_32&a_33 \end{vmatrix}$ and $A_y$ is Cofactor of $a_y$, then value of $\Delta$ is:

$\begin{array}{l} (A) \quad a_{11}A_{31}+a_{12}A_{32}+a_{13}A_{33} & (B) \quad a_{11}A_{11}+a_{12}A_{21}+a_{13}A_{31} \\(C) \quad a_{21}A_{11}+a_{22}A_{12}+a_{23}A_{13} & (D) \quad a_{11}A_{11}+a_{21}A_{21}+a_{31}A_{31} \\\end{array}$

Toolbox:
• $\bigtriangleup$ of a determinant is the sum of the product of the element of a column (or a row)with its corresponding cofactors.
We know that $\bigtriangleup$=Sum of the product of the element of a column (or a row) with its corresponding cofactors.
Hence $\bigtriangleup=a_{11}A_{11}+a_{21}A_{21}+a_{31}A_{31}$
Hence the correct answer is D.

edited Feb 28, 2013